Title: Self-similarity: two examples of singularity formation
Abstract: I will discuss the role that self-similarity plays in singularity formation in PDE's. I will illustrate it on two examples. The first is the role of self-similar solutions of the compressible Euler equations in the proof of a finite time blow up for the energy super-critical nonlinear Schrodinger and the compressible Navier-Stokes equations. The second example is in the context of the Einstein vacuum equations of general relativity and involves  a geometric notion of “twisted self-similarity” which plays an important role in the proof of existence of vacuum spacetimes with naked singularities.